On foliations of the projective space with a Kupka component
If \({\mathcal F}\) is a singular holomorphic foliation of codimension 1 on \(\mathbb {CP}(n)\), with \(n\geq 3\), a connected component \(K\) of the set of singularities of \({\mathcal F}\) is said to be a Kupka component (of \({\mathcal F}\)) if, for any \(p\in K\), there is a neighbourhood \(U\) around \(p\) such that the restriction \({\mathcal F}_{|U}\) is generated by a 1-form \(\omega\) verifying \(\omega(p)=0\) and \(d\omega(p)\neq 0\). Such a component \(K\) is smooth and of codimension 2 in \(\mathbb{CP}(n)\), and, along it, \({\mathcal F}\) is locally the product \({\mathcal F}_0\times \mathbb D^{n-2}\), where \({\mathcal F}_0\) is a singular foliation in a disk \(\mathbb D^2\). The foliation \({\mathcal F}_0\) is called the transversal type of \(K\), and after \textit{X. Gómez-Mont} and \textit{A. Lins-Neto} [Topology 30, No. 3, 315--334 (1991; Zbl 0735.57014)] one knows that it can be represented by a linear equation \(p w dz - q z dw=0\), with \(p,q\) positive coprime integers. When \(p=q=1\), \(K\) is said to be a radial Kupka component. The foliations with a non-radial Kupka component were classified by \textit{D. Cerveau} and \textit{A. Lins-Neto} in [Astérisque. 222, 93--133 (1994; Zbl 0823.32014)] and \textit{O. Calvo-Andrade} in [Bol. Soc. Bras. Mat., Nova Sér. 30, No. 2, 183--197 (1999; Zbl 1058.32023)]. They proved that \(K\) is a complete intersection, \(K=\{F=G=0\}\), and \({\mathcal F}\) is defined by a pencil of algebraic hypersurfaces \(pGdF - qFdG =0\) where \(p,q \in \mathbb Z^+\) and \(pq\geq 2\). Recently, \textit{O. Calvo-Andrade} (unpublished) has proved that this result can be extended to the radial case: If \({\mathcal F}\) is a singular holomorphic foliation of codimension 1 on \(\mathbb{CP}(n)\), with \(n\geq 3\), with a radial Kupka component \(K\), then \(K\) is a complete intersection, \(K=\{F=G=0\}\), and \({\mathcal F}\) is defined by a pencil of algebraic hypersurfaces \(GdF -FdG =0\). The main result of the paper under review is a very interesting method to obtain a global rational first integral for a holomorphic foliation of codimension 1 on \(\mathbb{CP}(n)\), with \(n\geq 3\) from the existence of an algebraic curve that does not contain singularities of the foliation and which admits a transversal projective structure of the foliation along it (Theorem 2). As a nice application of this result, the author obtains alternative proofs of the classifications for the foliations with a Kupka component that we have mentioned above. The paper is very well written and proves interesting and nice results concerning foliations of the projective space. It is worth reading even for nonspecialists.
scientific article; zbMATH DE number 5666403